Explanations for the anomalously high mobility of protons in liquid water began with Grotthuss's idea1, 2 of 'structural diffusion' nearly two centuries ago. Subsequent explanations have refined this concept by invoking thermal hopping3, 4, proton tunnelling5, 6 or solvation effects7. More recently, two main structural models have emerged for the hydrated proton. Eigen8, 9 proposed the formation of an H9O4+ complex in which an H3O+ core is strongly hydrogen-bonded to three H2O molecules. Zundel10, 11, meanwhile, supported the notion of an H5O2+ complex in which the proton isshared between two H2O molecules. Here we use ab initio path integral12, 13, 14 simulations to address this question. These simulations include time-independent equilibrium thermal and quantum fluctuations of all nuclei, and determine interatomic interactions from the electronic structure. We find that the hydrated proton forms a fluxional defect in the hydrogen-bonded network, with both H9O4+ and H5O2+ occurring only in thesense of 'limiting' or 'ideal' structures. The defect can become delocalized over several hydrogen bonds owing to quantum fluctuations. Solvent polarization induces a small barrier to proton transfer, which is washed out by zero-point motion. The proton can consequently be considered part of a 'low-barrier hydrogen bond'15, 16, in which tunnelling is negligible and the simplest concepts of transition-state theory do not apply. The rate of proton diffusion is determined by thermally induced hydrogen-bond breaking in the second solvation shell.